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SIDELIGHTS  ON  RELATIVITY 


BY     THE     SAME    AUTHOR 

RELATIVITY  :  THE  SPECIAL  AND 
THE  GENERAL  THEORY 


SIDELIGHTS  ON 
RELATIVITY 


BY 

ALBERT  EINSTEIN,  Ph.D. 

PROFESSOR  OF  PHYSICS  IN  THB  UNIVERSITY  OP  BBRLIN 


I.    ETHER  AND  RELATIVITY 
II.    GEOMETRY  AND  EXPERIENCE 


TRANSLATED  BY 
G.  B.  JEFFERY,  D.Sc.,  AND  W.  PERRETT,  PH.D. 


METHUEN  &  CO.  LTD. 

36  ESSEX    STREET   W.G. 

LONDON 


'•  '^r'  «L  STUDIES  - 

KXF LESLEY  PLACE 

TO  6,  CANADA, 


FEB271S32' 


This  Translation  was  first  published  in  1922 


ETHER  AND  THE  THEORY  OF 
RELATIVITY 

An  Address  delivered  on  May  5th,  1920, 
in  the  University  of  Leyden 


ETHER  AND  THE  THEORY  OF 
RELATIVITY 

HOW  does  it  come  about  that 
alongside  of  the  idea  of  ponderable 
matter,  which  is  derived  by  abstrac- 
tion from  everyday  life,  the  physicists 
set  the  idea  of  the  existence  of  another 
kind  of  matter,  the  ether  ?  The  explana- 
tion is  probably  to  be  sought  in  those 
phenomena  which  have  given  rise  to  the 
theory  of  action  at  a  distance,  and  in  the 
properties  of  light  which  have  led  to  the 
undulatory  theory.  Let  us  devote  a  little 
while  to  the  consideration  of  these  two 
subjects. 

Outside  of  physics  we  know  nothing 
of  action  at  a  distance.  When  we  try 
to  connect  cause  and  effect  in  the  experi- 
ences which  natural  objects  afford  us, 
it  seems  at  first  as  if  there  were  no  other 


.  E  5-3 


4      SIDELIGHTS  ON  RELATIVITY 

mutual  actions  than  those  of  immediate 
contact,  e.g.  the  communication  of  motion 
by  impact,  push  and  pull,  heating  or  induc- 
ing combustion  by  means  of  a  flame,  etc. 
It  is  true  that  even  in  everyday  experi- 
ence weight,  which  is  in  a  sense  action  at 
a  distance,  plays  a  very  important  part. 
But  since  in  daily  experience  the  weight 
of  bodies  meets  us  as  something  constant, 
something  not  linked  to  any  cause  which 
is  variable  in  time  or  place,  we  do  not 
in  everyday  life  speculate  as  to  the  cause 
of  gravity,  and  therefore  do  not  become 
conscious  of  its  character  as  action  at 
a  distance.  It  was  Newton's  theory  of 
gravitation  that  first  assigned  a  cause 
for  gravity  by  interpreting  it  as  action 
at  a  distance,  proceeding  from  masses. 
Newton's  theory  is  probably  the  greatest 
stride  ever  made  in  the  effort  towards 
the  causal  nexus  of  natural  phenomena. 
And  yet  this  theory  evoked  a  lively  sense 
of  discomfort  among  Newton's  contempor- 
aries, because  it  seemed  to  be  in  conflict 
with  the  principle  springing  from  the  rest 
of  experience,  that  there  can  be  reciprocal 


ETHER  AND   RELATIVITY         5 

action  only  through  contact,  and  not 
through  immediate  action  at  a  distance. 
It  is  only  with  reluctance  that  man's 
desire  for  knowledge  endures  a  dualism 
of  this  kind.  How  was  unity  to  be  pre- 
served in  his  comprehension  of  the  forces 
of  nature  ?  Either  by  trying  to  look 
upon  contact  forces  as  being  themselves 
distant  forces  which  admittedly  are  obser- 
vable only  at  a  very  small  distance— 
and  this  was  the  road  which  Newton's 
followers,  who  were  entirely  under  the 
spell  of  his  doctrine,  mostly  preferred 
to  take  ;  or  by  assuming  that  the  Newton- 
ian action  at  a  distance  is  only  apparently 
immediate  action  at  a  distance,  but  in 
truth  is  conveyed  by  a  medium  permeat- 
ing space,  whether  by  movements  or  by 
elastic  deformation  of  this  medium.  Thus 
the  endeavour  toward  a  unified  view  of 
the  nature  of  forces  leads  to  the  hypothesis 
of  an  ether.  This  hypothesis,  to  be  sure, 
did  not  at  first  bring  with  it  any  advance 
in  the  theory  of  gravitation  or  in  physics 
generally,  so  that  it  became  customary 
to  treat  Newton's  law  of  force  as  an  axiom 


6      SIDELIGHTS  ON   RELATIVITY 

not  further  reducible.  But  the  ether  hypo- 
thesis was  bound  always  to  play  some 
part  in  physical  science,  even  if  at  first 
only  a  latent  part. 

When  in  the  first  half  of  the  nineteenth 
century  the  far-reaching  similarity  was  re- 
vealed which  subsists  between  the  properties 
of  light  and  those  of  elastic  waves  in  pon- 
derable bodies,  the  ether  hypothesis  found 
fresh  support.  It  appeared  beyond  question 
that  light  must  be  interpreted  as  a  vibratory 
process  in  an  elastic,  inert  medium  filling 
up  universal  space.  It  also  seemed  to 
be  a  necessary  consequence  of  the  fact 
that  light  is  capable  of  polarisation  that 
this  medium,  the  ether,  must  be  of  the 
nature  of  a  solid  body,  because  transverse 
waves  are  not  possible  in  a  fluid,  but 
only  in  a  solid.  Thus  the  physicists  were 
bound  to  arrive  at  the  theory  of  the  "  quasi- 
rigid  "  luminiferous  ether,  the  parts  of 
which  can  carry  out  no  movements 
relatively  to  one  another  except  the  small 
movements  of  deformation  which  corre- 
spond to  light-waves. 

This  theory — also  called  the  theory  of 


ETHER  AND  RELATIVITY         7 

the  stationary  luminiferous  ether— more- 
over found  a  strong  support  in  an  experi- 
ment which  is  also  of  fundamental  import- 
ance in  the  special  theory  of  relativity, 
the  experiment  of  Fizeau,  from  which 
one  was  obliged  to  infer  that  the  lumini- 
ferous ether  does  not  take  part  in  the 
movements  of  bodies.  The  phenomenon 
of  aberration  also  favoured  the  theory 
of  the  quasi-rigid  ether. 

The  development  of  the  theory  of  electric- 
ity along  the  path  opened  up  by  Maxwell 
and  Lorentz  gave  the  development  of  our 
ideas  concerning  the  ether  quite  a  peculiar 
and  unexpected  turn.  For  Maxwell  him- 
self the  ether  indeed  still  had  properties 
which  were  purely  mechanical,  although 
of  a  much  more  complicated  kind  than 
the  mechanical  properties  of  tangible  solid 
bodies.  But  neither  Maxwell  nor  his 
followers  succeeded  in  elaborating  a 
mechanical  model  for  the  ether  which 
might  furnish  a  satisfactory  mechanical 
interpretation  of  Maxwell's  laws  of  the 
electro-magnetic  field.  The  laws  were  clear 
and  simple,  the  mechanical  interpretations 


8      SIDELIGHTS  ON   RELATIVITY 

clumsy  and  contradictory.  Almost  imper- 
ceptibly the  theoretical  physicists  adapted 
themselves  to  a  situation  which,  from  the 
standpoint  of  their  mechanical  programme, 
was  very  depressing.  They  were  parti- 
cularly influenced  by  the  electro-dynamical 
investigations  of  Heinrich  Hertz.  For 
whereas  they  previously  had  required  of 
a  conclusive  theory  that  it  should  con- 
tent itself  with  the  fundamental  concepts 
which  belong  exclusively  to  mechanics 
(e.g.  densities,  velocities,  deformations, 
stresses)  they  gradually  accustomed  them- 
selves to  admitting  electric  and  magnetic 
force  as  fundamental  concepts  side  by 
side  with  those  of  mechanics,  without 
requiring  a  mechanical  interpretation  for 
them.  Thus  the  purely  mechanical  view 
of  nature  was  gradually  abandoned.  But 
this  change  led  to  a  fundamental  dualism 
which  in  the  long-run  was  insupportable. 
A  way  of  escape  was  now  sought  in  the 
reverse  direction,  by  reducing  the  principles 
of  mechanics  to  those  of  electricity,  and 
this  especially  as  confidence  in  the  strict 
validity  of  the  equations  of  Newton's 


ETHER  AND  RELATIVITY        9 

mechanics  was  shaken  by  the  experi- 
ments with  /3-rays  and  rapid  kathode 
rays. 

This  dualism  still  confronts  us  in  unex- 
tenuated form  in  the  theory  of  Hertz, 
where  matter  appears  not  only  as  the 
bearer  of  velocities,  kinetic  energy,  and 
mechanical  pressures,  but  also  as  the 
bearer  of  electromagnetic  fields.  Since 
such  fields  also  occur  in  vacuo — i.e.  in  free 
ether — the  ether  also  appears  as  bearer  of 
electromagnetic  fields.  The  ether  appears 
indistinguishable  in  its  functions  from 
ordinary  matter.  Within  matter  it  takes 
part  in  the  motion  of  matter  and  in  empty 
space  it  has  everywhere  a  velocity ;  so 
that  the  ether  has  a  definitely  assigned 
velocity  throughout  the  whole  of  space. 
There  is  no  fundamental  difference  between 
Hertz's  ether  and  ponderable  matter  (which 
in  part  subsists  in  the  ether). 

The  Hertz  theory  suffered  not  only 
from  the  defect  of  ascribing  to  matter  and 
ether,  on  the  one  hand  mechanical  states, 
and  on  the  other  hand  electrical  states, 
which  do  not  stand  in  any  conceivable 


io    SIDELIGHTS  ON  RELATIVITY 

relation  to  each  other  ;  it  was  also  at  vari- 
ance with  the  result  of  Fizeau's  important 
experiment  on  the  velocity  of  the  propaga- 
tion of  light  in  moving  fluids,  and  with 
other  established  experimental  results. 

Such  was  the  state  of  things  when  H.  A. 
Lorentz  entered  upon  the  scene.  He  brought 
theory  into  harmony  with  experience  by 
means  of  a  wonderful  simplification  of 
theoretical  principles.  He  achieved  this, 
the  most  important  advance  in  the  theory 
of  electricity  since  Maxwell,  by  taking 
from  ether  its  mechanical,  and  from  matter 
its  electromagnetic  qualities.  As  in  empty 
space,  so  too  in  the  interior  of  material 
bodies,  the  ether,  and  not  matter  viewed 
atomistically,  was  exclusively  the  seat 
of  electromagnetic  fields.  According  to 
Lorentz  the  elementary  particles  of  matter 
alone  are  capable  of  carrying  out  move- 
ments ;  their  electromagnetic  activity  is 
entirely  confined  to  the  carrying  of  electric 
charges.  Thus  Lorentz  succeeded  in  re- 
ducing all  electromagnetic  happenings  to 
Maxwell's  equations  for  free  space. 

As   to    the    mechanical   nature    of    the 


ETHER  AND  RELATIVITY       n 

Lorentzian  ether,  it  may  be  said  of  it, 
in  a  somewhat  playful  spirit,  that  im- 
mobility is  the  only  mechanical  property 
of  which  it  has  not  been  deprived  by 
H.  A.  Lorentz.  It  may  be  added  that  the 
whole  change  in  the  conception  of  the 
ether  which  the  special  theory  of  relativity 
brought  about,  consisted  in  taking  away 
from  the  ether  its  last  mechanical  quality, 
namely,  its  immobility.  How  this  is  to 
be  understood  will  forthwith  be  expounded. 
The  space-time  theory  and  the  kine- 
matics of  the  special  theory  of  relativity 
were  modelled  on  the  Maxwell-Lorentz 
theory  of  the  electromagnetic  field.  This 
theory  therefore  satisfies  the  conditions 
of  the  special  theory  of  relativity,  but 
when  viewed  from  the  latter  it  acquires 
a  novel  aspect.  For  if  K  be  a  system 
of  co-ordinates  relatively  to  which  the 
Lorentzian  ether  is  at  rest,  the  Maxwell- 
Lorentz  equations  are  valid  primarily  with 
reference  to  K.  But  by  the  special  theory 
of  relativity  the  same  equations  with- 
out any  change  of  meaning  also  hold  in 
relation  to  any  new  system  of  co-ordinates 


12     SIDELIGHTS   ON   RELATIVITY 

K'  which  is  moving  in  uniform  trans- 
lation relatively  to  K.  Now  comes  the 
anxious  question  : — Why  must  I  in  the 
theory  distinguish  the  K  system  above  all 
K'  systems,  which  are  physically  equivalent 
to  it  in  all  respects,  by  assuming  that  the 
ether  is  at  rest  relatively  to  the  K  system  ? 
For  the  theoretician  such  an  asymmetry 
in  the  theoretical  structure,  with  no  cor- 
responding asymmetry  in  the  system  of 
experience,  is  intolerable.  If  we  assume 
the  ether  to  be  at  rest  relatively  to  K, 
but  in  motion  relatively  to  K',  the  physical 
equivalence  of  K  and  K'  seems  to  me 
from  the  logical  standpoint,  not  indeed 
downright  incorrect,  but  nevertheless 
inacceptable. 

The  next  position  which  it  was  possible 
to  take  up  in  face  of  this  state  of  things 
appeared  to  be  the  following.  The  ether 
does  not  exist  at  all.  The  electromag- 
netic fields  are  not  states  of  a  medium, 
and  are  not  bound  down  to  any  bearer, 
but  they  are  independent  realities  which 
are  not  reducible  to  anything  else,  exactly 
like  the  atoms  of  ponderable  matter.  This 


ETHER  AND   RELATIVITY       13 

conception  suggests  itself  the  more  readily 
as,  according  to  Lorentz's  theory,  electro- 
magnetic radiation,  like  ponderable  matter, 
brings  impulse  and  energy  with  it,  and 
as,  according  to  the  special  theory  of 
relativity,  both  matter  and  radiation  are 
but  special  forms  of  distributed  energy, 
ponderable  mass  losing  its  isolation  and 
appearing  as  a  special  form  of  energy. 

More  careful  reflection  teaches  us,  how- 
ever, that  the  special  theory  of  relativity 
does  not  compel  us  to  deny  ether.  We 
may  assume  the  existence  of  an  ether  ; 
only  we  must  give  up  ascribing  a  definite 
state  of  motion  to  it,  i.e.  we  must  by 
abstraction  take  from  it  the  last  mechan- 
ical characteristic  which  Lorentz  had  still 
left  it.  We  shall  see  later  that  this  point 
of  view,  the  conceivability  of  which  I 
shall  at  once  endeavour  to  make  more 
intelligible  by  a  somewhat  halting  com- 
parison, is  justified  by  the  results  of 
the  general  theory  of  relativity. 

Think  of  waves  on  the  surface  of  water. 
Here  we  can  describe  two  entirely  differ- 
ent things.  Either  we  may  observe  how 


14    SIDELIGHTS  ON   RELATIVITY 

the  undulatory  surface  forming  the  bound- 
ary between  water  and  air  alters  in  the 
course  of  time ;  or  else— with  the  help 
of  small  floats,  for  instance — we  can 
observe  how  the  position  of  the  separate 
particles  of  water  alters  in  the  course 
of  time.  If  the  existence  of  such  floats 
for  tracking  the  motion  of  the  particles 
of  a  fluid  were  a  fundamental  impos- 
sibility in  physics — if,  in  fact,  nothing 
else  whatever  were  observable  than  the 
shape  of  the  space  occupied  by  the  water 
as  it  varies  in  time,  we  should  have  no 
ground  for  the  assumption  that  water 
consists  of  movable  particles.  But  all 
the  same  we  could  characterise  it  as  a 
medium. 

We  have  something  like  this  in  the 
electromagnetic  field.  For  we  may  pic- 
ture the  field  to  ourselves  as  consisting 
of  lines  of  force.  If  we  wish  to  inter- 
pret these  lines  of  force  to  ourselves  as 
something  material  in  the  ordinary  sense, 
we  are  tempted  to  interpret  the  dynamic 
processes  as  motions  of  these  lines  of 
force,  such  that  each  separate  line  of 


ETHER  AND  RELATIVITY       15 

force  is  tracked  through  the  course  of 
time.  It  is  well  known,  however,  that 
this  way  of  regarding  the  electromagnetic 
field  leads  to  contradictions. 

Generalising  we  must  say  this : — There 
may  be  supposed  to  be  extended  physical 
objects  to  which  the  idea  of  motion  can- 
not be  applied.  They  may  not  be  thought 
of  as  consisting  of  particles  which  allow 
themselves  to  be  separately  tracked  through 
time.  In  Minkowski's  idiom  this  is  ex- 
pressed as  follows  : — Not  every  extended 
conformation  in  the  four-dimensional  world 
can  be  regarded  as  composed  of  world- 
threads.  The  special  theory  of  relativity 
forbids  us  to  assume  the  ether  to  consist 
of  particles  observable  through  time,  but 
the  hypothesis  of  ether  in  itself  is  not 
in  conflict  with  the  special  theory  of 
relativity.  Only  we  must  be  on  our  guard 
against  ascribing  a  state  of  motion  to  the 
ether. 

Certainly,  from  the  standpoint  of  the 
special  theory  of  relativity,  the  ether 
hypothesis  appears  at  first  to  be  an  empty 
hypothesis.  In  the  equations  of  the 


16    SIDELIGHTS  ON   RELATIVITY 

electromagnetic  field  there  occur,  in  addition 
to  the  densities  of  the  electric  charge, 
only  the  intensities  of  the  field.  The 
career  of  electromagnetic  processes  in- vacua 
appears  to  be  completely  determined  by 
these  equations,  uninfluenced  by  other  phy- 
sical quantities.  The  electromagnetic  fields 
appear  as  ultimate,  irreducible  realities, 
and  at  first  it  seems  superfluous  to  postu- 
late a  homogeneous,  isotropic  ether- 
medium,  and  to  envisage  electromagnetic 
fields  as  states  of  this  medium. 

But  on  the  other  hand  there  is  a  weighty 
argument  to  be  adduced  in  favour  of  the 
ether  hypothesis.  To  deny  the  ether  is 
ultimately  to  assume  that  empty  space 
has  no  physical  qualities  whatever.  The 
fundamental  facts  of  mechanics  do  not 
harmonize  with  this  view.  For  the  me- 
chanical behaviour  of  a  corporeal  system 
hovering  freely  in  empty  space  depends 
not  only  on  relative  positions  (distances) 
and  relative  velocities,  but  also  on  its 
state  of  rotation,  which  physically  may 
be  taken  as  a  characteristic  not  apper- 
taining to  the  system  in  itself.  In  order 


ETHER  AND  RELATIVITY       17 

to  be  able  to  look  upon  the  rotation  of 
the  system,  at  least  formally,  as  some- 
thing real,  Newton  objectivises  space. 
Since  he  classes  his  absolute  space  together 
with  real  things,  for  him  rotation  relative 
to  an  absolute  space  is  also  something 
real.  Newton  might  no  less  well  have 
called  his  absolute  space  "  Ether  "  ;  what 
is  essential  is  merely  that  besides  observ- 
able objects,  another  thing,  which  is 
not  perceptible,  must  be  looked  upon  as 
real,  to  enable  acceleration  or  rotation 
to  be  looked  upon  as  something  real. 

It  is  true  that  Mach  tried  to  avoid 
having  to  accept  as  real  something  which 
is  not  observable  by  endeavouring  to 
substitute  in  mechanics  a  mean  acceler- 
ation with  reference  to  the  totality  of  the 
masses  in  the  universe  in  place  of  an 
acceleration  with  reference  to  absolute 
space.  But  inertial  resistance  opposed  to 
relative  acceleration  of  distant  masses 
presupposes  action  at  a  distance ;  and 
as  the  modern  physicist  does  not  believe 
that  he  may  accept  this  action  at  a  dis- 
tance, he  comes  back  once  more,  if  he 

2 


i8    SIDELIGHTS  ON   RELATIVITY 

follows  Mach,  to  the  ether,  which  has 
to  serve  as  medium  for  the  effects  of 
inertia.  But  this  conception  of  the  ether 
to  which  we  are  led  by  Mach's  way  of 
thinking  differs  essentially  from  the  ether 
as  conceived  by  Newton,  by  Fresnel,  and 
by  Lorentz.  Mach's  ether  not  only  con- 
ditions the  behaviour  of  inert  masses, 
but  is  also  conditioned  in  its  state  by 
them. 

Mach's  idea  finds  its  full  development  in 
the  ether  of  the  general  theory  of  relativity. 
According  to  this  theory  the  metrical 
qualities  of  the  continuum  of  space-time 
differ  in  the  environment  of  different  points 
of  space-time,  and  are  partly  conditioned 
by  the  matter  existing  outside  of  the 
territory  under  consideration.  This  space- 
time  variability  of  the  reciprocal  relations 
of  the  standards  of  space  and  time,  or, 
perhaps,  the  recognition  of  the  fact  that 
"  empty  space  "  in  its  physical  relation  is 
neither  homogeneous  nor  isotropic,  compel- 
ling us  to  describe  its  state  by  ten  functions 
(the  gravitation  potentials  gMV),  has,  I 
think,  finally  disposed  of  the  view  that 


ETHER  AND  RELATIVITY       19 

space  is  physically  empty.  But  therewith 
the  conception  of  the  ether  has  again 
acquired  an  intelligible  content,  although 
this  content  differs  widely  from  that  of  the 
ether  of  the  mechanical  undulatory  theory 
of  light.  The  ether  of  the  general  theory 
of  relativity  is  a  medium  which  is  itself 
devoid  of  all  mechanical  and  kinematic^! 
qualities,  but  helps  to  determine  mechanical 
(and  electromagnetic)  events. 

What  is  fundamentally  new  in  the  ether 
of  the  general  theory  of  relativity  as  op- 
posed to  the  ether  of  Lorentz  consists  in 
this,  that  the  state  of  the  former  is  at 
every  place  determined  by  connections  with 
the  matter  and  the  state  of  the  ether  in 
neighbouring  places,  which  are  amenable 
to  law  in  the  form  of  differential  equations  ; 
whereas  the  state  of  the  Lorentzian  ether 
in  the  absence  of  electromagnetic  fields  is 
conditioned  by  nothing  outside  itself,  and 
is  everywhere  the  same.  The  ether  of 
the  general  theory  of  relativity  is  trans- 
muted conceptually  into  the  ether  of 
Lorentz  if  we  substitute  constants  for  the 
functions  of  space  which  describe  the 


20    SIDELIGHTS  ON  RELATIVITY 

former,  disregarding  the  causes  which  con- 
dition its  state.  Thus  we  may  also  say, 
I  think,  that  the  ether  of  the  general  theory 
of  relativity  is  the  outcome  of  the  Lorent- 
zian  ether,  through  relativation. 

As  to  the  part  which  the  new  ether  is 
to  play  in  the  physics  of  the  future  we  are 
not  yet  clear.  We  know  that  it  determines 
the  metrical  relations  in  the  space-time 
continuum,  e.g.  the  configurative  possi- 
bilities of  solid  bodies  as  well  as  the  gravita- 
tional fields  ;  but  we  do  not  know  whether 
it  has  an  essential  share  in  the  structure  of 
the  electrical  elementary  particles  consti- 
tuting matter.  Nor  do  we  know  whether 
it  is  only  in  the  proximity  of  ponderable 
masses  that  its  structure  differs  essentially 
from  that  of  the  Lorentzian  ether  ;  whether 
the  geometry  of  spaces  of  cosmic  extent 
is  approximately  Euclidean.  But  we  can 
assert  by  reason  of  the  relativistic  equations 
of  gravitation  that  there  must  be  a  depart- 
ure from  Euclidean  relations,  with  spaces 
of  cosmic  order  of  magnitude,  if  there 
exists  a  positive  mean  density,  no  matter 
how  small,  of  the  matter  in  the  universe. 


ETHER  AND   RELATIVITY       21 

In  this  case  the  universe  must  of  necessity 
be  spatially  unbounded  and  of  finite  magni- 
tude, its  magnitude  being  determined  by 
the  value  of  that  mean  density. 

If  we  consider  the  gravitational  field  and 
the  electromagnetic  field  from  the  stand- 
point of  the  ether  hypothesis,  we  find  a  re- 
markable difference  between  the  two.  There 
can  be  no  space  nor  any  part  of  space  without 
gravitational  potentials ;  for  these  confer 
upon  space  its  metrical  qualities,  without 
which  it  cannot  be  imagined  at  all.  The 
existence  of  the  gravitational  field  is 
inseparably  bound  up  with  the  existence 
of  space.  On  the  other  hand  a  part  of 
space  may  very  well  be  imagined  without 
an  electromagnetic  field ;  thus  in  contrast 
with  the  gravitational  field,  the  electro- 
magnetic field  seems  to  be  only  secondarily 
linked  to  the  ether,  the  formal  nature  of 
the  electromagnetic  field  being  as  yet  in 
no  way  determined  by  that  of  gravitational 
ether.  From  the  present  state  of  theory 
it  looks  as  if  the  electromagnetic  field,  as 
opposed  to  the  gravitational  field,  rests  upon 
an  entirely  new  formal  motif,  as  though 


22    SIDELIGHTS  ON   RELATIVITY 

nature  might  just  as  well  have  endowed  the 
gravitational  ether  with  fields  of  quite 
another  type,  for  example,  with  fields  of  a 
scalar  potential,  instead  of  fields  of  the 
electromagnetic  type. 

Since  according  to  our  present  concep- 
tions the  elementary  particles  of  matter 
are  also,  in  their  essence,  nothing  else  than 
condensations  of  the  electromagnetic  field, 
our  present  view  of  the  universe  presents 
two  realities  which  are  completely  separated 
from  each  other  conceptually,  although 
connected  causally,  namely,  gravitational 
ether  and  electromagnetic  field,  or — as 
they  might  also  be  called — space  and 
matter. 

Of  course  it  would  be  a  great  advance  if 
we  could  succeed  in  comprehending  the 
gravitational  field  and  the  electromagnetic 
field  together  as  one  unified  conformation. 
Then  for  the  first  time  the  epoch  of  theor- 
etical physics  founded  by  Faraday  and 
Maxwell  would  reach  a  satisfactory  con- 
clusion. The  contrast  between  ether  and 
matter  would  fade  away,  and,  through  the 
general  theory  of  relativity,  the  whole  of 


ETHER  AND   RELATIVITY       23 

physics  would  become  a  complete  system  of 
thought,  like  geometry,  kinematics,  and  the 
theory  of  gravitation.  An  exceedingly 
ingenious  attempt  in  this  direction  has 
been  made  by  the  mathematician  H.  Weyl ; 
but  I  do  not  believe  that  his  theory 
will  hold  its  ground  in  relation  to  reality. 
Further,  in  contemplating  the  immediate 
future  of  theoretical  physics  we  ought  not 
unconditionally  to  reject  the  possibility 
that  the  facts  comprised  in  the  quantum 
theory  may  set  bounds  to  the  field  theory 
beyond  which  it  cannot  pass. 

Recapitulating,  we  may  say  that  accord- 
ing to  the  general  theory  of  relativity  space 
is  endowed  with  physical  qualities  ;  in  this 
sense,  therefore,  there  exists  an  ether. 
According  to  the  general  theory  of  relativity 
space  without  ether  is  unthinkable ;  for 
in  such  space  there  not  only  would  be  no 
propagation  of  light,  but  also  no  possibility 
of  existence  for  standards  of  space  and 
time  (measuring-rods  and  clocks),  nor  there- 
fore any  space-time  intervals  in  the  physical 
sense.  But  this  ether  may  not  be  thought 
of  as  endowed  with  the  quality  characteris- 


24    SIDELIGHTS  ON  RELATIVITY 

tic  of  ponderable  media,  as  consisting  of 
parts  which  may  be  tracked  through  time. 
The  idea  of  motion  may  not  be  applied 
to  it. 


GEOMETRY  AND  EXPERIENCE 

An  expanded  form  of  an  Address  to 
the  Prussian  Academy  of  Sciences 
in  Berlin  on  January  27th,  1921. 


GEOMETRY    AND    EXPERIENCE 

ONE  reason  why  mathematics  enjoys 
special  esteem,  above  all  other 
sciences,  is  that  its  laws  are  ab- 
solutely certain  and  indisputable,  while 
those  of  all  other  sciences  are  to  some 
extent  debatable  and  in  constant  danger 
of  being  overthrown  by  newly  discovered 
facts.  In  spite  of  this,  the  investigator  in 
another  department  of  science  would  not 
need  to  envy  the  mathematician  if  the 
laws  of  mathematics  referred  to  objects  of 
our  mere  imagination,  and  not  to  objects 
of  reality.  For  it  cannot  occasion  sur- 
prise that  different  persons  should  arrive 
at  the  same  logical  conclusions  when 
they  have  already  agreed  upon  the  funda- 
mental laws  (axioms),  as  well  as  the 
methods  by  which  other  laws  are  to  be 
deduced  therefrom.  But  there  is  another 

27 


28    SIDELIGHTS  ON   RELATIVITY 

reason  for  the  high  repute  of  mathematics, 
in  that  it  is  mathematics  which  affords  the 
exact  natural  sciences  a  certain  measure 
of  security,  to  which  without  mathematics 
they  could  not  attain. 

At  this  point  an  enigma  presents  itself 
which  in  all  ages  has  agitated  inquiring 
minds.  How  can  it  be  that  mathematics, 
being  after  all  a  product  of  human  thought 
which  is  independent  of  experience,  is  so 
admirably  appropriate  to  the  objects  of 
reality?  Is  human  reason,  then,  without 
experience,  merely  by  taking  thought,  able 
to  fathom  the  properties  of  real  things. 

In  my  opinion  the  answer  to  this  question 
is,  briefly,  this : — As  far  as  the  laws  of 
mathematics  refer  to  reality,  they  are 
not  certain ;  and  as  far  as  they  are  certain, 
they  do  not  refer  to  reality.  It  seems 
to  me  that  complete  clearness  as  to  this 
state  of  things  first  became  common 
property  through  that  new  departure  in 
mathematics  which  is  known  by  the  name 
of  mathematical  logic  or  "  Axiomatics." 
The  progress  achieved  by  axiomatics  consists 
in  its  having  neatly  separated  the  logical- 


GEOMETRY  AND  EXPERIENCE  29 

formal  from  its  objective  or  intuitive  con- 
tent; according  to  axiomatics  the  logical- 
formal  alone  forms  the  subject-matter  of 
mathematics,  which  is  not  concerned  with 
the  intuitive  or  other  content  associated 
with  the  logical-formal. 

Let  us  for  a  moment  consider  from  this 
point  of  view  any  axiom  of  geometry,  for 
instance,  the  following : — Through  two 
points  in  space  there  always  passes  one 
and  only  one  straight  line.  How  is  this 
axiom  to  be  interpreted  in  the  older  sense 
and  in  the  more  modern  sense  ? 

The  older  interpretation  : — Every  one 
knows  what  a  straight  line  is,  and  what  a 
point  is.  Whether  this  knowledge  springs 
from  an  ability  of  the  human  mind  or  from 
experience,  from  some  collaboration  of 
the  two  or  from  some  other  source,  is  not 
for  the  mathematician  to  decide.  He 
leaves  the  question  to  the  philosopher. 
Being  based  upon  this  knowledge,  which 
precedes  all  mathematics,  the  axiom  stated 
above  is,  like  all  other  axioms,  self-evident, 
that  is,  it  is  the  expression  of  a  part  of 
this  d  priori  knowledge. 


30    SIDELIGHTS  ON   RELATIVITY 

The  more  modern  interpretation  :— 
Geometry  treats  of  entities  which  are 
denoted  by  the  words  straight  line,  point, 
etc.  These  entities  do  not  take  for  granted 
any  knowledge  or  intuition  whatever,  but 
they  presuppose  only  the  validity  of  the 
axioms,  such  as  the  one  stated  above, 
which  are  to  be  taken  in  a  purely  formal 
sense,  i.e.  as  void  of  all  content  of  intuition 
or  experience.  These  axioms  are  free 
creations  of  the  human  mind.  All  other 
propositions  of  geometry  are  logical  infer- 
ences from  the  axioms  (which  are  to  be 
taken  in  the  nominalistic  sense  only). 
The  matter  of  which  geometry  treats  is 
first  defined  by  the  axioms.  Schlick  in 
his  book  on  epistemology  has  therefore 
characterised  axioms  very  aptly  as 
"  implicit  definitions." 

This  view  of  axioms,  advocated  by 
modern  axiomatics,  purges  mathematics 
of  all  extraneous  elements,  and  thus  dispels 
the  mystic  obscurity  which  formerly  sur- 
rounded the  principles  of  mathematics. 
But  a  presentation  of  its  principles  thus 
clarified  makes  it  also  evident  that  mathe- 


GEOMETRY  AND  EXPERIENCE  31 

matics  as  such  cannot  predicate  anything 
about  perceptual  objects  or  real  objects. 
In  axiomatic  geometry  the  words  "  point/' 
"  straight  line/'  etc.,  stand  only  for  empty 
conceptual  schemata.  That  which  gives 
them  substance  is  not  relevant  to  mathe- 
matics. 

Yet  on  the  other  hand  it  is  certain  that 
mathematics  generally,  and  particularly 
geometry,  owes  its  existence  to  the  need 
which  was  felt  of  learning  something  about 
the  relations  of  real  things  to  one  another. 
The  very  word  geometry,  which,  of  course, 
means  earth-measuring,  proves  this.  For 
earth-measuring  has  to  do  with  the  possi- 
bilities of  the  disposition  of  certain  natural 
objects  with  respect  to  one  another,  namely, 
with  parts  of  the  earth,  measuring-lines, 
measuring-wands,  etc.  It  is  clear  that 
the  system  of  concepts  of  axiomatic 
geometry  alone  cannot  make  any  assertions 
as  to  the  relations  of  real  objects  of  this 
kind,  which  we  will  call  practically-rigid 
bodies.  To  be  able  to  make  such  asser- 
tions, geometry  must  be  stripped  of  its 
merely  logical-formal  character  by  the 


32     SIDELIGHTS  ON   RELATIVITY 

co-ordination  of  real  objects  of  experience 
with  the  empty  conceptual  frame-work  of 
axiomatic  geometry.  To  accomplish  this, 
we  need  only  add  the  proposition  : — Solid 
bodies  are  related,  with  respect  to  their 
possible  dispositions,  as  are  bodies  in 
Euclidean  geometry  of  three  dimensions. 
Then  the  propositions  of  Euclid  contain 
affirmations  as  to  the  relations  of  practi- 
cally-rigid bodies. 

Geometry  thus  completed  is  evidently  a 
natural  science  ;  we  may  in  fact  regard  it 
as  the  most  ancient  branch  of  physics. 
Its  affirmations  rest  essentially  on 
induction  from  experience,  but  not  on 
logical  inferences  only.  We  will  call  this 
completed  geometry  "  practical  geometry/' 
and  shall  distinguish  it  in  what  follows 
from  "  purely  axiomatic  geometry."  The 
question  whether  the  practical  geometry 
of  the  universe  is  Euclidean  or  not  has  a 
clear  meaning,  and  its  answer  can  only 
be  furnished  by  experience.  All  linear 
measurement  in  physics  is  practical  geo- 
metry in  this  sense,  so  too  is  geodetic 
and  astronomical  linear  measurement,  if 


GEOMETRY  AND  EXPERIENCE  33 

we  call  to  our  help  the  law  of  experience 
that  light  is  propagated  in  a  straight  line, 
and  indeed  in  a  straight  line  in  the  sense 
of  practical  geometry. 

I  attach  special  importance  to  the  view 
of  geometry  which  I  have  just  set  forth, 
because  without  it  I  should  have  been 
unable  to  formulate  the  theory  of  rela- 
tivity. Without  it  the  following  reflection 
would  have  been  impossible  : — -In  a  system 
of  reference  rotating  relatively  to  an  inert 
system,  the  laws  of  disposition  of  rigid 
bodies  do  not  correspond  to  the  rules  of 
Euclidean  geometry  on  account  of  the 
Lorentz  contraction ;  thus  if  we  admit 
non-inert  systems  we  must  abandon  Eucli- 
dean geometry.  The  decisive  step  in  the 
transition  to  general  co-variant  equations 
would  certainly  not  have  been  taken  if 
the  above  interpretation  had  not  served 
as  a  stepping-stone.  If  we  deny  the 
relation  between  the  body  of  axiomatic 
Euclidean  geometry  and  the  practically- 
rigid  body  of  reality,  we  readily  arrive  at 
the  following  view,  which  was  entertained 
by  that  acute  and  profound  thinker, 

3 


34    SIDELIGHTS  ON   RELATIVITY 

H.  Poincare  : — Euclidean  geometry  is  dis- 
tinguished above  all  other  imaginable 
axiomatic  geometries  by  its  simplicity. 
Now  since  axiomatic  geometry  by  itself 
contains  no  assertions  as  to  the  reality 
which  can  be  experienced,  but  can  do  so 
only  in  combination  with  physical  laws, 
it  should  be  possible  and  reasonable — 
whatever  may  be  the  nature  of  reality — to 
retain  Euclidean  geometry.  For  if  con- 
tradictions between  theory  and  experience 
manifest  themselves,  we  should  rather 
decide  to  change  physical  laws  than  to 
change  axiomatic  Euclidean  geometry.  If 
we  deny  the  relation  between  the  practi- 
cally-rigid body  and  geometry,  we  shall 
indeed  not  easily  free  ourselves  from  the 
convention  that  Euclidean  geometry  is  to 
be  retained  as  the  simplest.  Why  is  the 
equivalence  of  the  practically-rigid  body 
and  the  body  of  geometry — which  suggests 
itself  so  readily — denied  by  Poincare  and 
other  investigators  ?  Simply  because  under 
closer  inspection  the  real  solid  bodies  in 
nature  are  not  rigid,  because  their  geo- 
metrical behaviour,  that  is,  their  possi- 


GEOMETRY  AND  EXPERIENCE  35 

bilities  of  relative  disposition,  depend  upon 
temperature,  external  forces,  etc.  Thus 
the  original,  immediate  relation  between 
geometry  and  physical  reality  appears 
destroyed,  and  we  feel  impelled  toward 
the  following  more  general  view,  which 
characterizes  Poincare's  standpoint. 
Geometry  (G)  predicates  nothing  about 
the  relations  of  real  things,  but  only 
geometry  together  with  the  purport  (P) 
of  physical  laws  can  do  so.  Using  symbols, 
we  may  say  that  only  the  sum  of  (G)  -f 
(P)  is  subject  to  the  control  of  experience. 
Thus  (G)  may  be  chosen  arbitrarily,  and 
also  parts  of  (P)  ;  all  these  laws  are  con- 
ventions. All  that  is  necessary  to  avoid 
contradictions  is  to  choose  the  remainder 
of  (P)  so  that  (G)  and  the  whole  of  (P) 
are  together  in  accord  with  experience. 
Envisaged  in  this  way,  axiomatic  geometry 
and  the  part  of  natural  law  which  has  been 
given  a  conventional  status  appear  as 
epistemologically  equivalent. 

Sub  specie  aeterni  Poincare,  in  my 
opinion,  is  right.  The  idea  of  the 
measuring-rod  and  the  idea  of  the  clock 


36    SIDELIGHTS  ON  RELATIVITY 

co-ordinated  with  it  in  the  theory  of 
relativity  do  not  find  their  exact  corre- 
spondence in  the  real  world.  It  is  also 
clear  that  the  solid  body  and  the  clock  do 
not  in  the  conceptual  edifice  of  physics 
play  the  part  of  irreducible  elements,  but 
that  of  composite  structures,  which  may 
not  play  any  independent  part  in  theo- 
retical physics.  But  it  is  my  conviction 
that  in  the  present  stage  of  development 
of  theoretical  physics  these  ideas  must 
still  be  employed  as  independent  ideas ; 
for  we  are  still  far  from  possessing  such 
certain  knowledge  of  theoretical  principles 
as  to  be  able  to  give  exact  theoretical 
constructions  of  solid  bodies  and  clocks. 

Further,  as  to  the  objection  that  there 
are  no  really  rigid  bodies  in  nature,  and 
that  therefore  the  properties  predicated  of 
rigid  bodies  do  not  apply  to  physical 
reality,— this  objection  is  by  no  means  so 
radical  as  might  appear  from  a  hasty 
examination.  For  it  is  not  a  difficult  task 
to  determine  the  physical  state  of  a  measur- 
ing-rod so  accurately  that  its  behaviour 
relatively  to  other  measuring-bodies  shall 


GEOMETRY  AND  EXPERIENCE  37 

be  sufficiently  free  from  ambiguity  to  allow 
it  to  be  substituted  for  the  "  rigid  "  body. 
It  is  to  measuring-bodies  of  this  kind  that 
statements  as  to  rigid  bodies  must  be 
referred. 

All  practical  geometry  is  based  upon  a 
principle  which  is  accessible  to  experience, 
and  which  we  will  now  try  to  realise.  We 
will  call  that  which  is  enclosed  between  two 
boundaries,  marked  upon  a  practically- 
rigid  body,  a  tract.  We  imagine  two 
practically-rigid  bodies,  each  with  a  tract 
marked  out  on  it.  These  two  tracts  are 
said  to  be  "  equal  to  one  another  "  if  the 
boundaries  of  the  one  tract  can  be  brought 
to  coincide  permanently  with  the  bound- 
aries of  the  other.  We  now  assume 
that : 

If  two  tracts  are  found  to  be  equal 
once  and  anywhere,  they  are  equal  always 
and  everywhere. 

Not  only  the  practical  geometry  of 
Euclid,  but  also  its  nearest  generalisation, 
the  practical  geometry  of  Riemann,  and 
therewith  the  general  theory  of  rela- 
tivity, rest  upon  this  assumption.  Of  the 


38    SIDELIGHTS  ON   RELATIVITY 

experimental  reasons  which  warrant  this 
assumption  I  will  mention  only  one.  The 
phenomenon  of  the  propagation  of  light 
in  empty  space  assigns  a  tract,  namely, 
the  appropriate  path  of  light,  to  each 
interval  of  local  time,  and  conversely. 
Thence  it  follows  that  the  above  assump- 
tion for  tracts  must  also  hold  good  for 
intervals  of  clock-time  in  the  theory  of 
relativity.  Consequently  it  may  be  formu- 
lated as  follows : — If  two  ideal  clocks 
are  going  at  the  same  rate  at  any,  time 
and  at  any  place  (being  then  in  immedi- 
ate proximity  to  each  other),  they  will 
always  go  at  the  same  rate,  no  matter 
where  and  when  they  are  again  compared 
with  each  other  at  one  place. — If  this 
law  were  not  valid  for  real  clocks,  the 
proper  frequencies  for  the  separate  atoms 
of  the  same  chemical  element  would  not 
be  in  such  exact  agreement  as  experience 
demonstrates.  The  existence  of  sharp 
spectral  lines  is  a  convincing  experimental 
proof  of  the  above-mentioned  principle 
of  practical  geometry.  This  is  the  ulti- 
mate foundation  in  fact  which  enables 


GEOMETRY  AND  EXPERIENCE  39 

us  to  speak  with  meaning  of  the  mensura- 
tion, in  Riemann's  sense  of  the  word, 
of  the  four-dimensional  continuum  of 
space-time. 

The  question  whether  the  structure  of 
this  continuum  is  Euclidean,  or  in  accord- 
ance with  Riemann's  general  scheme,  or 
otherwise,  is,  according  to  the  view  which 
is  here  being  advocated,  properly  speaking  a 
physical  question  which  must  be  answered 
by  experience,  and  not  a  question  of  a 
mere  convention  to  be  selected  on  prac- 
tical grounds.  Riemann's  geometry  will 
be  the  right  thing  if  the  laws  of  disposi- 
tion of  practically-rigid  bodies  are  trans- 
formable into  those  of  the  bodies  of 
Euclid's  geometry  with  an  exactitude 
which  increases  in  proportion  as  the  dimen- 
sions of  the  part  of  space-time  under 
consideration  are  diminished. 

It  is  true  that  this  proposed  physical 
interpretation  of  geometry  breaks  down 
when  applied  immediately  to  spaces  of 
sub-molecular  order  of  magnitude.  But 
nevertheless,  even  in  questions  as  to  the 
constitution  of  elementary  particles,  it 


40     SIDELIGHTS  ON   RELATIVITY 

retains  part  of  its  importance.  For  even 
when  it  is  a  question  of  describing  the 
electrical  elementary  particles  constitut- 
ing matter,  the  attempt  may  still  be 
made  to  ascribe  physical  importance  to 
those  ideas  of  fields  which  have  been 
physically  defined  for  the  purpose  of  des- 
cribing the  geometrical  behaviour  of  bodies 
which  are  large  as  compared  with  the 
molecule.  Success  alone  can  decide  as 
to  the  justification  of  such  an  attempt, 
which  postulates  physical  reality  for  the 
fundamental  principles  of  Riemann's  geo- 
metry outside  of  the  domain  of  their 
physical  definitions.  It  might  possibly 
turn  out  that  this  extrapolation  has  no 
better  warrant  than  the  extrapolation  of 
the  idea  of  temperature  to  parts  of  a  body 
of  molecular  order  of  magnitude. 

It  appears  less  problematical  to  extend 
the  ideas  of  practical  geometry  to  spaces 
of  cosmic  order  of  magnitude.  It  might, 
of  course,  be  objected  that  a  construc- 
tion composed  of  solid  rods  departs  more 
and  more  from  ideal  rigidity  in  propor- 
tion as  its  spatial  extent  becomes  greater. 


GEOMETRY  AND  EXPERIENCE    41 

But  it  will  hardly  be  possible,  I  think, 
to  assign  fundamental  significance  to  this 
objection.  Therefore  the  question  whether 
the  universe  is  spatially  finite  or  not 
seems  to  me  decidedly  a  pregnant  ques- 
tion in  the  sense  of  practical  geometry. 
I  do  not  even  consider  it  impossible  that 
this  question  will  be  answered  before 
long  by  astronomy.  Let  us  call  to  mind 
what  the  general  theory  of  relativity 
teaches  in  this  respect.  It  offers  two 
possibilities  : — 

1.  The  universe  is  spatially  infinite.    This 
can   be    so    only   if   the    average  spatial 
density  of  the  matter  in  universal  space, 
concentrated  in    the    stars,  vanishes,  i.e. 
if  the  ratio  of  the  total  mass  of  the  stars 
to   the  magnitude   of  the   space   through 
which    they  are    scattered    approximates 
indefinitely  to  the  value  zero  when  the 
spaces  taken  into  consideration  are  con- 
stantly greater  and  greater. 

2.  The  universe  is  spatially  finite.    This 
must  be  so,  if  there  is  a  mean  density 
of    the    ponderable    matter    in    universal 
space   differing   from   zero.     The   smaller 


42   .SIDELIGHTS  ON   RELATIVITY 

that   mean   density,    the   greater   is    the 
volume  of  universal  space. 

I  must  not  fail  to  mention  that  a  theo- 
retical argument  can  be  adduced  in  favour 
of  the  hypothesis  of  a  finite  universe. 
The  general  theory  of  relativity  teaches 
that  the  inertia  of  a  given  body  is 
greater  as  there  are  more  ponderable 
masses  in  proximity  to  it ;  thus  it  seems 
very  natural  to  reduce  the  total  effect 
of  inertia  of  a  body  to  action  and  reac- 
tion between  it  and  the  other  bodies  in 
the  universe,  as  indeed,  ever  since  New- 
ton's time,  gravity  has  been  completely 
reduced  to  action  and  reaction  between 
bodies.  From  the  equations  of  the  general 
theory  of  relativity  it  can  be  deduced  that 
this  total  reduction  of  inertia  to  reciprocal 
action  between  masses — as  required  by 
E.  Mach,  for  example — is  possible  only  if 
the  universe  is  spatially  finite. 

On  many  physicists  and  astronomers 
this  argument  makes  no  impression. 
Experience  alone  can  finally  decide  which 
of  the  two  possibilities  is  realised  in 
nature.  How  can  experience  furnish  an 


GEOMETRY  AND  EXPERIENCE.    43 

answer  ?     At  first  it  might  seem  possible 
to  determine  the  mean  density  of  matter 
by  observation  of  that  part  of  the  universe 
which    is    accessible    to    our    perception. 
This   hope   is   illusory.     The    distribution 
of  the  visible  stars  is  extremely  irregular, 
so  that  we  on  no  account  may  venture 
to  set  down  the  mean  density  of  star- 
matter  in  the  universe   as  equal,   let  us 
say,  to  the  mean  density  in  the  Milky 
Way.     In   any   case,   however   great   the 
space   examined  may  be,   we   could  not 
feel  convinced  that  there  were  no  more 
stars    beyond    that    space.     So    it    seems 
impossible  to  estimate  the  mean  density. 
But  there  is  another  road,  which  seems 
to  me  more  practicable,  although  it  also 
presents  great  difficulties.     For  if  we  inquire 
into  the  deviations  shown  by  the  conse- 
quences of  the  general  theory  of  relativity 
which  are  accessible  to  experience,  when 
these  are  compared  with  the  consequences 
of  the  Newtonian  theory,  we  first  of  all 
find    a    deviation    which    shows   itself   in 
close  proximity  to  gravitating  mass,  and 
has  been  confirmed  in  the  case  of  the  planet 


44    SIDELIGHTS  ON  RELATIVITY 

Mercury.  But  if  the  universe  is  spatially 
finite  there  is  a  second  deviation  from 
the  Newtonian  theory,  which,  in  the  lan- 
guage of  the  Newtonian  theory,  may  be 
expressed  thus  : — The  gravitational  field 
is  in  its  nature  such  as  if  it  were  produced, 
not  only  by  the  ponderable  masses,  but 
also  by  a  mass-density  of  negative  sign, 
distributed  uniformly  throughout  space. 
Since  this  factitious  mass-density  would 
have  to  be  enormously  small,  it  could 
make  its  presence  felt  only  in  gravitating 
systems  of  very  great  extent. 

Assuming  that  we  know,  let  us  say, 
the  statistical  distribution  of  the  stars 
in  the  Milky  Way,  as  well  as  their  masses, 
then  by  Newton's  law  we  can  calculate 
the  gravitational  field  and  the  mean 
velocities  which  the  stars  must  have,  so 
that  the  Milky  Way  should  not  collapse 
under  the  mutual  attraction  of  its  stars, 
but  should  maintain  its  actual  extent. 
Now  if  the  actual  velocities  of  the  stars, 
which  can,  of  course,  be  measured,  were 
smaller  than  the  calculated  velocities,  we 
should  have  a  proof  that  the  actual  attrac- 


GEOMETRY  AND  EXPERIENCE  45 

tions  at  great  distances  are  smaller  than 
by  Newton's  law.  From  such  a  deviation 
it  could  be  proved  indirectly  that  the 
universe  is  finite.  It  would  even  be  possible 
to  estimate  its  spatial  magnitude. 

Can  we  picture  to  ourselves  a  three- 
dimensional  universe  which  is  finite,  yet 
unbounded  ? 

The  usual  answer  to  this  question  is 
"  No,"  but  that  is  not  the  right  answer. 
The  purpose  of  the  following  remarks  is 
to  show  that  the  answer  should  be  "  Yes." 
I  want  to  show  that  without  any  extra- 
ordinary difficulty  we  can  illustrate  the 
theory  of  a  finite  universe  by  means  of  a 
mental  image  to  which,  with  some  practice, 
we  shall  soon  grow  accustomed. 

First  of  all,  an  obervation  of  episte- 
mological  nature.  A  geometrical-physical 
theory  as  such  is  incapable  of  being  directly 
pictured,  being  merely  a  system  of  concepts. 
But  these  concepts  serve  the  purpose  of 
bringing  a  multiplicity  of  real  or  imaginary 
sensory  experiences  into  connection  in  the 
mind.  To  "  visualise  "  a  theory,  or  bring 
it  home  to  one's  mind,  therefore  means  to 


46    SIDELIGHTS  ON   RELATIVITY 

give  a  representation  to  that  abundance 
of  experiences  for  which  the  theory  supplies 
the  schematic  arrangement.  In  the  present 
case  we  have  to  ask  ourselves  how  we  can 
represent  that  relation  of  solid  bodies  with 
respect  to  their  reciprocal  disposition  (con- 
tact) which  corresponds  to  the  theory  of 
a  finite  universe.  There  is  really  nothing 
new  in  what  I  have  to  say  about  this ; 
but  innumerable  questions  addressed  to 
me  prove  that  the  requirements  of  those 
who  thirst  for  knowledge  of  these  matters 
have  not  yet  been  completely  satisfied. 
So,  will  the  initiated  please  pardon  me, 
if  part  of  what  I  shall  bring  forward  has 
long  been  known  ? 

What  do  we  wish  to  express  when  we 
say  that  our  space  is  infinite  ?  Nothing 
more  than  that  we  might  lay  any  number 
whatever  of  bodies  of  equal  sizes  side  by 
side  without  ever  filling  space.  Suppose 
that  we  are  provided  with  a  great  many 
wooden  cubes  all  of  the  same  size.  In 
accordance  with  Euclidean  geometry  we 
can  place  them  above,  beside,  and  behind 
one  another  so  as  to  fill  a  part  of  space  of 


GEOMETRY  AND  EXPERIENCE  47 

any  dimensions ;  but  this  construction 
would  never  be  finished ;  we  could  go 
on  adding  more  and  more  cubes  without 
ever  finding  that  there  was  no  more  room. 
That  is  what  we  wish  to  express  when  we 
say  that  space  is  infinite.  It  would  be 
better  to  say  that  space  is  infinite  in 
relation  to  practically-rigid  bodies,  assum- 
ing that  the  laws  of  disposition  for  these 
bodies  are  given  by  Euclidean  geometry. 

Another  example  of  an  infinite  continuum 
is  the  plane.  On  a  plane  surface  we  may 
lay  squares  of  cardboard  so  that  each  side 
of  any  square  has  the  side  of  another 
square  adjacent  to  it.  The  construction 
is  never  finished ;  we  can  always  go  on 
laying  squares — if  their  laws  of  disposition 
correspond  to  those  of  plane  figures  of 
Euclidean  geometry.  The  plane  is  there- 
fore infinite  in  relation  to  the  cardboard 
squares.  Accordingly  we  say  that  the 
plane  is  an  infinite  continuum  of  two 
dimensions,  and  space  an  infinite  con- 
tinuum of  three  dimensions.  What  is  here 
meant  by  the  number  of  dimensions,  I 
think  I  may  assume  to  be  known. 


48    SIDELIGHTS  ON   RELATIVITY 

Now  we  take  an  example  of  a  two- 
dimensional  continuum  which  is  finite, 
but  unbounded.  We  imagine  the  surface 
of  a  large  globe  and  a  quantity  of  small 
paper  discs,  all  of  the  same  size.  We 
place  one  of  the  discs  anywhere  on  the 
surface  of  the  globe.  If  we  move  the  disc 
about,  anywhere  we  like,  on  the  surface 
of  the  globe,  we  do  not  come  upon  a 
limit  or  boundary  anywhere  on  the  journey. 
Therefore  we  say  that  the  spherical  surface 
of  the  globe  is  an  unbounded  continuum. 
Moreover,  the  spherical  surface  is  a  finite 
continuum.  For  if  we  stick  the  paper 
discs  on  the  globe,  so  that  no  disc  overlaps 
another,  the  surface  of  the  globe  will 
finally  become  so  full  that  there  is  no 
room  for  another  disc.  This  simply  means 
that  the  spherical  surface  of  the  globe 
is  finite  in  relation  to  the  paper  discs. 
Further,  the  spherical  surface  is  a  non- 
Euclidean  continuum  of  two  dimensions, 
that  is  to  say,  the  laws  of  disposition 
for  the  rigid  figures  lying  in  it  do  not 
agree  with  those  of  the  Euclidean  plane. 
This  can  be  shown  in  the  following  way. 


GEOMETRY  AND  EXPERIENCE  49 

Place  a  paper  disc  on  the  spherical  surface, 
and  around  it  in  a  circle  place  six  more 
discs,  each  of  which  is  to  be  surrounded 
in  turn  by  six  discs,  and  so  on.  If  this 
construction  is  made  on  a  plane  surface, 
we  have  an  uninterrupted  disposition  in 
which  there  are  six  discs  touching  every 
disc  except  those  which  lie  on  the  outside. 


COJL 


FIG.  i. 

On  the  spherical  surface  the  construction 
also  seems  to  promise  success  at  the  outset, 
and  the  smaller  the  radius  of  the  disc 
in  proportion  to  that  of  the  sphere,  the 
more  promising  it  seems.  But  as  the 
construction  progresses  it  becomes  more 
and  more  patent  that  the  disposition  of 
the  discs  in  the  manner  indicated,  without 
interruption,  is  not  possible,  as  it  should 
be  possible  by  Euclidean  geometry  of  the 
the  plane  surface.  In  this  way  creatures 

4 


50    SIDELIGHTS  ON   RELATIVITY 

which  cannot  leave  the  spherical  surface, 
and  cannot  even  peep  out  from  the  spherical 
surface  into  three-dimensional  space,  might 
discover,  merely  by  experimenting  with 
discs,  that  their  two-dimensional  "  space  " 
is  not  Euclidean,  but  spherical  space. 

From  the  latest  results  of  the  theory 
of  relativity  it  is  probable  that  our  three- 
dimensional  space  is  also  approximately 
spherical,  that  is,  that  the  laws  of  dis- 
position of  rigid  bodies  in  it  are  not  given 
by  Euclidean  geometry,  but  approximately 
by  spherical  geometry,  if  only  we  consider 
parts  of  space  which  are  sufficiently  great. 
Now  this  is  the  place  where  the  reader's 
imagination  boggles.  "  Nobody  can  imagine 
this  thing,"  he  cries  indignantly.  "  It  can 
be  said,  but  cannot  be  thought.  I  can 
represent  to  myself  a  spherical  surface 
well  enough,  but  nothing  analogous  to  it 
in  three  dimensions." 

We  must  try  to  surmount  this  barrier 
in  the  mind,  and  the  patient  reader  will 
see  that  it  is  by  no  means  a  particularly 
difficult  task.  For  this  purpose  we  will 
first  give  our  attention  once  more  to 


GEOMETRY  AND  EXPERIENCE  51 

the  geometry  of  two-dimensional  spherical 
surfaces.  In  the  adjoining  figure  let  K 
be  the  spherical  surface,  touched  at  5 
by  a  plane,  E,  which,  for  facility  of  pre- 
sentation, is  shown  in  the  drawing  as  a 
bounded  surface.  Let  L  be  a  disc  on  the 
spherical  surface.  Now  let  us  imagine 
that  at  the  point  N  of  the  spherical  surface, 


FIG.  2. 

diametrically  opposite  to  S,  there  is  a 
luminous  point,  throwing  a  shadow  L' 
of  the  disc  L  upon  the  plane  E.  Every 
point  on  the  sphere  has  its  shadow  on 
the  plane.  If  the  disc  on  the  sphere  K 
is  moved,  its  shadow  L'  on  the  plane  E 
also  moves.  When  the  disc  L  is  at  5,  it 
almost  exactly  coincides  with  its  shadow. 
If  it  moves  on  the  spherical  surface  away 


52     SIDELIGHTS  ON   RELATIVITY 

from  S  upwards,  the  disc  shadow  Lr  on 
the  plane  also  moves  away  from  S  on  the 
plane  outwards,  growing  bigger  and  bigger. 
As  the  disc  L  approaches  the  luminous 
point  N,  the  shadow  moves  off  to  infinity, 
and  becomes  infinitely  great. 

Now  we  put  the  question,  What  are 
the  laws  of  disposition  of  the  disc-shadows 
L'  on  the  plane  E?  Evidently  they  are 
exactly  the  same  as  the  laws  of  disposition 
of  the  discs  L  on  the  spherical  surface. 
For  to  each  original  figure  on  K  there 
is  a  corresponding  shadow  figure  on  E. 
If  two  discs  on  K  are  touching,  their 
shadows  on  E  also  touch.  The  shadow- 
geometry  on  the  plane  agrees  with  the 
the  disc-geometry  on  the  sphere.  If  we 
call  the  disc-shadows  rigid  figures,  then 
spherical  geometry  holds  good  on  the  plane 
E  with  respect  to  these  rigid  figures.  More- 
over, the  plane  is  finite  with  respect  to  the 
disc-shadows,  since  only  a  finite  number 
of  the  shadows  can  find  room  on  the 
plane. 

At  this  point  somebody  will  say,  "  That 
is  nonsense.  The  disc-shadows  are  not 


GEOMETRY  AND  EXPERIENCE  53 

rigid  figures.  We  have  only  to  move  a 
two-foot  rule  about  on  the  plane  E  to  con- 
vince ourselves  that  the  shadows  constantly 
increase  in  size  as  they  move  away  from 
S  on  the  plane  towards  infinity."  But 
what  if  the  two-foot  rule  were  to  behave 
on  the  plane  E  in  the  same  way  as  the 
disc-shadows  L'  ?  It  would  then  be  impos- 
sible to  show  that  the  shadows  increase 
in  size  as  they  move  away  from  S  ;  such  an 
assertion  would  then  no  longer  have  any 
meaning  whatever.  In  fact  the  only  ob- 
jective assertion  that  can  be  made  about 
the  disc-shadows  is  just  this,  that  they 
are  related  in  exactly  the  same  way  as 
are  the  rigid  discs  on  the  spherical  surface 
in  the  sense  of  Euclidean  geometry. 

We  must  carefully  bear  in  mind  that 
our  statement  as  to  the  growth  of  the 
disc-shadows,  as  they  move  away  from 
S  towards  infinity,  has  in  itself  no  objec- 
tive meaning,  as  long  as  we  are  unable 
to  employ  Euclidean  rigid  bodies  which 
can  be  moved  about  on  the  plane  E  for 
the  purpose  of  comparing  the  size  of  the 
disc-shadows.  In  respect  of  the  laws  of 


54    SIDELIGHTS   ON   RELATIVITY 

disposition  of  the  shadows  L' ,  the  point 
5  has  no  special  privileges  on  the  plane 
any  more  than  on  the  spherical  surface. 

The  representation  given  above  of  spheri- 
cal geometry  on  the  plane  is  important 
for  us,  because  it  readily  allows  itself  to 
be  transferred  to  the  three-dimensional 
case. 

Let  us  imagine  a  point  S  of  our  space, 
and  a  great  number  of  small  spheres,  L', 
which  can  all  be  brought  to  coincide  with 
one  another.  But  these  spheres  are  not 
to  be  rigid  in  the  sense  of  Euclidean  geo- 
metry ;  their  radius  is  to  increase  (in 
the  sense  of  Euclidean  geometry)  when  they 
are  moved  away  from  S  towards  infinity, 
and  this  increase  is  to  take  place  in  exact 
accordance  with  the  same  law  as  applies 
to  the  increase  of  the  radii  of  the  disc- 
shadows  L'  on  the  plane. 

After  having  gained  a  vivid  mental 
image  of  the  geometrical  behaviour  of  our 
L'  spheres,  let  us  assume  that  in  our  space 
there  are  no  rigid  bodies  at  all  in  the  sense 
of  Euclidean  geometry,  but  only  bodies 
having  the  behaviour  of  our  L'  spheres. 


GEOMETRY  AND  EXPERIENCE  55 

Then  we  shall  have  a  vivid  representation 
of  three-dimensional  spherical  space,  or, 
rather  of  three-dimensional  spherical  geo- 
metry. Here  our  spheres  must  be  called 
"  rigid "  spheres.  Their  increase  in  size 
as  they  depart  from  S  is  not  to  be  detected 
by  measuring  with  measuring-rods,  any 
more  than  in  the  case  of  the  disc-shadows 
on  E,  because  the  standards  of  measure- 
ment behave  in  the  same  way  as  the 
spjieres.  Space  is  homogeneous,  that  is 
to  say,  the  same  spherical  configurations 
are  possible  in  the  environment  of  all 
points.1  Our  space  is  finite,  because, 
in  consequence  of  the  "  growth  "  of  the 
spheres,  only  a  finite  number  of  them 
can  find  room  in  space. 

In  this  way,  by  using  as  stepping-stones 
the  practice  in  thinking  and  visualisation 
which  Euclidean  geometry  gives  us,  we 
have  acquired  a  mental  picture  of  spheri- 
cal geometry.  We  may  without  difficulty 

1  This  is  intelligible  without  calculation — but 
only  for  the  two-dimensional  case — if  we  revert  once 
more  to  the  case  of  the  disc  on  the  surface  of  the 
sphere. 


56    SIDELIGHTS  ON   RELATIVITY 

impart  more  depth  and  vigour  to  these 
ideas  by  carrying  out  special  imaginary 
constructions.  Nor  would  it  be  difficult 
to  represent  the  case  of  what  is  called 
elliptical  geometry  in  an  analogous  manner. 
My  only  aim  to-day  has  been  to  show 
that  the  human  faculty  of  visualisation  is 
by  no  means  bound  to  capitulate  to  non- 
Euclidean  geometry. 


Printed  in  Great  Britain  by 
Butler  &  Tanner, 
Frame  and  London 


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